Solve for $x$ : $ 2|x + 9| + 10 = 5|x + 9| + 6 $
Subtract $ {2|x + 9|} $ from both sides: $ \begin{eqnarray} 2|x + 9| + 10 &=& 5|x + 9| + 6 \\ \\ {- 2|x + 9|} && {- 2|x + 9|} \\ \\ 10 &=& 3|x + 9| + 6 \end{eqnarray} $ Subtract $6$ from both sides: $ \begin{eqnarray} 10 &=& 3|x + 9| + 6 \\ \\ {- 6} && {- 6} \\ \\ 4 &=& 3|x + 9| \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{4} {{3}} = \dfrac{3|x + 9|} {{3}} $ Simplify: $ \dfrac{4}{3} = |x + 9| $ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ -\dfrac{4}{3} = x + 9 $ or $ \dfrac{4}{3} = x + 9 $ Solve for the solution where $x + 9$ is negative: $ - \dfrac{4}{3} = x + 9$ Subtract ${9}$ from both sides: $ \begin{eqnarray} - \dfrac{4}{3} &=& x + 9 \\ \\ {- 9} && {- 9} \\ \\ -\dfrac{4}{3} - 9 &=& x \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $3$ $ - \dfrac{4}{3} {- \dfrac{27}{3}} = x $ $ -\dfrac{31}{3} = x $ Then calculate the solution where $x + 9$ is positive: $ \dfrac{4}{3} = x + 9 $ Subtract ${9}$ from both sides: $ \begin{eqnarray} \dfrac{4}{3} &=& x + 9 \\ \\ {- 9} && {- 9} \\ \\ \dfrac{4}{3} - 9 &=& x \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $3$ $ \dfrac{4}{3} {- \dfrac{27}{3}} = x $ $ -\dfrac{23}{3} = x $ Thus, the correct answer is $x = -\dfrac{31}{3} $ or $x = -\dfrac{23}{3} $.